his-togram with 200×200 bins over the range [0, π/2)×[0.0,0.3] and normalized so as ∫ ∫
P(θ, Q)dθdQ= 1. At R/Rcr = 20.0,21.0,22.0, the enstrophy takes
0.3
00 1
θ Q
(a) (b)
(c) (d) (e)
50
0
Fig. 2.12: Joint probability density functions of the enstrophy Q and the angle θ. The horizontal axis is the angle θ, the vertical axis is the enstrophy Q and the contour is the joint PDF at R/Rcr = (a) 20.0, (b) 21.0, (c) 22.0, (d) 23.0, (e) 24.0.
both large and small values at large angles. However, at R/Rcr = 23.0,24.0, it is observed that the enstrophy does not take large values at large angles.
It may be worth noting that the change of the relation between the angle and the enstrophy takes place at the Reynolds number close to that of the hyperbolic-nonhyperbolic transition of the attractor.
2.6 Discussions and Conclutions
In the chaotic Kolmogorov flows, Inubushiet al. [52] observed the hyperbolic-nonhyperbolic transition (in §IV. of [52]) employing the covariant Lyapunov analysis [36]. Therefore In this chapter, we focused our attention on the relations between the hyperbolic and physical properties.
We studied the correlation decay of vorticity for several Reynolds numbers across the hyperbolic-nonhyperbolic transition point. In lower-dimensional dynamical systems hyperbolic/nonhyperbolic properties are known to be re-lated to decay of correlations, especially, nonhyperbolicity usually leads to non-exponential decay of correlations. We found that the qualitative change
2 Relations between hyperbolic properties and physical properties of chaotic Kolmogorov flow
0 30 60 90 0
0.1 0.2 0.3 0.4 0.5
0 30 60 90 0
0.5 1 1.5
(a) (b) 2
Fig. 2.13: Joint probability density functions of the angleθand (a) the energy dissipation rate E and (b) energyE.
of the long-time correlation function occurs at the Reynolds number close to the hyperbolic-nonhyperbolic transition point and to the Reynolds num-ber where the 2nd positive Lyapunov exponent emerge, suggesting that the asymptotic decay of the time-correlation reflects the transition to nonhy-perbolicity and/or the emergence of “unstable mode” of the flow. Also, we reported that the angleθ is relevant to the enstrophyQ(the energy dissipa-tion rateε) ; the enstrophy is small when the angle is large, which holds at Reynolds numbers where the attractor is nonhyperbolic. A similar relation between the angle and the energy dissipation rate is also observed in GOY shell model (Kobayashi and Yamada [57]). They studied GOY model em-ploying the covariant Lyapunov analysis and found that the angle between the stable and unstable manifoldsθ is related to the energy dissipation rate in a similar manner (i.e. the angle θ tends to be small when the energy dissipation rate is large). Interestingly, this relation can not hold between the angle θ and the energy (in GOY model, the energy is not necessary cor-related to energy dissipation rate as in fully developed turbulence). It will be intriguing future work to see how these properties relate to each other and whether this relation holds in general dissipative systems including fully developed Navier-Stokes turbulence.
Chapter 3
Orbital instability of the
regeneration cycle in minimal Couette turbulence
3.1 Introduciton
As one of ‘generic’ properties of near-wall turbulence, a scaling law (known as Prandtl wall law) is observed in near wall region of a wide variety of wall turbulence such as turbulence in pipes, channels, ducts, and boundary layers, where a scaled mean velocity profile ¯u(z) is logarithmic: ¯u(z)∝logz (z is a scaled distance from the wall) 1. A number of researchers have studied this statistical property, and flow structures (so-called coherent structures) have been recognized as key elements to understand near-wall turbulence (Jim´enez and Moin [58], Hamilton et al. [59]). In order to find out mechanisms pro-ducing the wall turbulence, they searched numerically the minimal size of periodic box (minimal flow unit) in which we can observe the turbulence. As a result, in the minimal flow units, they found regeneration cycle consisting of breakdown and reformation of the coherent structures such as streamwise vortices and streaks which are high/low speed regions2 in Poiseuille turbu-lence [58] and in Couette turbuturbu-lence [59]. The regeneration cycle has been observed in many types of turbulence (Panton [60]) and was recently observed
1The streamwise mean flow profile scales with the kinetic viscosity ν and the wall friction velocity uτ, where the wall friction velocity is uτ =
√
ν⟨|∂U∂zx|⟩wall (the bracket
⟨·⟩wall denotes long-time and horizontal direction spatial mean at the walls andUxis the streamwise mean velocity. ).
2See Fig.3.5 and description of it for details.
3 Orbital instability of the regeneration cycle in minimal Couette turbulence
(a) (b)
(d) (c)
Fig. 3.1: Perturbation streamwise vorticity ωx for sinuous streak instability mode of the model streak at (a) αx = 0, (b) αx = π/2, (c) αx = π, and (d)αx = 3π/2 shown in Fig.9 in Schoppa and Hussain [71], whereα denotes the streamwise number. Positive and negative ωx are shown as solid and dotted contours respectively, and the bell-shaped line denotes the phase speed contour U = σi/α, where σi denotes the imaginary part of the eigenvalue.
The shading shows the regions of induced spanwise flow (in the direction of the thick arrow).
in experiments of boundary layer turbulence by Duriezet al. [61].
In order to describe the regeneration cycle, Hamilton et al. [59] and Waleffe [62] proposed a mechanisms (what they call self-sustaining process) which consists of streak instability, regeneration of the streamwise vortices, and formation of the streaks, by modeling the streaks and the streamwise vortices. On the streak instability, Schoppa and Hussain [71] investigated linear stability of models of the streaks numerically and found that these models are linearly unstable to sinuous instability mode (Fig.3.1. See also Figure 9 in [71]) which causes meandering of the straight streak as observed by Hamiltonet al. [59]. Linear stability of a corrugated vortex sheet, which is an inviscid model of the streak, is studied by Kawaharaet al.[72]. They found the vortex sheet is linearly unstable equally to both sinuous and varicose disturbances (i.e. their growth rates are identical) in a long-wave limit and
3.1 Introduciton
(a) (b)
Fig. 3.2: Unstable fundamental eigenstructures of a corrugated vortex sheet shown in Figure 3 (c,d) in Kawahara [72] for (a) sinuous mode and (b) vari-cose mode. The streamwise circulation density (see [72] for details) in the perturbed vortex sheet is shown for ξ0 = 1/3π where ξ0 denotes positions of the sheet. Red is positive (clockwise) and blue is negative (counterclockwise).
The disturbance velocity vectors, in a frame of reference moving with the real part of the phase velocity, are shown in the plane x= 0. One wavelength is shown both in the x- and in thez-directions.
discussed similarities between the obtained sinuous eigenfunction (Fig.3.2.
See also Figure 3 in [72]) and the invariant solutions of the Poiseuille flows and the Couette flows. There are numerous studies on linear stability of model streaks including the above models (see [72] and references therein) and most of them suggest that the sinuous mode is the most unstable (often referred to as the most ‘dangerous’) perturbations. Characteristics of the sinuous instability modes are (A) appearances of different signs streamwise vorticity alternatively, (B) localizations of streamwise vorticity near the low-speed streak ‘crest’ and the high-speed ‘trough’ regions (see Fig.3.1 and Fig.3.2).
However these models are not solutions of the full Navier-Stokes equation and it is unclear how the linear stability analyses of these models of steady solutions are crucial for understanding of the stability of the streak in the actual turbulent flows.
Following the meanderings of the straight streaks, the flow changes into fully three-dimensional turbulence, and streamwise vortices are expected to be generated. Toward an understanding of this process, many mechanisms has been proposed such as Waleffe [62] and Jim´enez and Moin [58] (See
Kawa-3 Orbital instability of the regeneration cycle in minimal Couette turbulence
hara [69] for review of regeneration mechanisms of streamwise vortices). Once the streamwise vortices are generated by some sort of mechanism, these vor-tices advect the gradient of the streamwise velocity in the cross-streamwise plane, which forms the streak structures. In other words, the streamwise vortices lift up low-velocity fluid from the bottom wall, and lifted down high-velocity fluid from the top wall. Kawahara [69] showed that an analytical model of the streamwise vortex forms the streak structures by the above mechanism. The formation of the streaks closes the regeneration cycle. Wal-effe [62] derived a low-dimensional model for understanding of the regenera-tion cycle (self-sustaining process) from the viewpoint of dynamical system theory, which has been modified and used to study transitions to turbulence over a wide parameter region (Kim and Moehlis [75]). While these descrip-tions and models are suggestive, the mechanisms composing the regeneration cycle, particularly the generation mechanism of the streamwise vortices, re-main unclear. Moreover, the whole of the regeneration cycle is expected to be understood not on the basis of the models and the phenomenological arguments but on the full Navier-Stokes equation.
One of the crucial steps toward understanding of the regeneration cycle on the basis of the full Navier-Stokes equation is finding of the UPO by Kawahara and Kida [25] which approximates turbulent statistics very well as mentioned in §1. Also, they found that temporal variations of spatial structures along the UPO exhibit the regeneration cycle. Recently, a lot of invariant solutions of the full Navier-Stokes equation and the (homoclinic and heteroclinic) connections between them have been found numerically and used to clarify the state space structures for understanding mechanisms of transition to turbulence and the regeneration cycle (see §1 for the brief review and Kawahara [29] for the detailed review).
We here focus our attention on the properties of the orbital instabilities of minimal Couette turbulence employing the covariant Lyapunov analysis, by which we can study ‘linear stability’ of the streaks in actual turbulence instead of the model streaks. Moreover, the covariant Lyapunov analysis is expected to capture not only the streak instability but also the other expo-nential instabilities in the whole of the regeneration cycle. Understanding of the instabilities of the cycle can be useful for a control of turbulence as well (Kawahara [22]). Also, some fundamental information on the attrac-tor can be obtained by the analysis such as the attracattrac-tor dimension3 and
3The attractor dimension of the minimal Couette turbulence is considered to be not high since the regeneration cycle can be characterized by only two coherent structures (i.e.
the streak and the streamwise vortex) and dynamics of several low-dimensional models
3.2 Couette flow system and numerical method